A continuation approach to state and adjoint calculation in optimal control applied to the reentry problem Knut Graichen Nicolas Petit Centre Automatique et Syst` emes, Ecole des Mines de Paris, 75272 Paris, France (e-mail: { knut.graichen , nicolas.petit }@ensmp.fr) Abstract: A well–known problem in indirect optimal control is to find a suitable initial guess for the adjoint states which is sufficiently close to the optimal solution. This paper presents a new homotopy approach to overcome this problem by deriving an auxiliary optimal control problem (OCP) for which the adjoint states are simply zero. A continuation method is employed to smoothly reach the original OCP. The auxiliary OCP can be derived with respect to any given initial trajectory of the system, for instance obtained by forward integration. The approach is applied to the space shuttle reentry problem, which represents a benchmark problem in optimal control due to its high numerical sensitivity with respect to the initial solution. Keywords: Optimal control; Homotopy continuation method; Aerospace applications. 1. INTRODUCTION Lately, optimal control problems (OCPs) have received much attention especially in the context of Model Predic- tive Control, which in turn has spurred interest in efficient numerical methods for real–time applications (Allg¨ower et al., 1999; Kouvaritakis and Cannon, 2001; Diehl et al., 2002). The numerical methods for solving OCPs can roughly be divided in two different classes. In the direct approach, the model equations of the considered system are discretized and the state and control trajectories are parametrized in order to obtain a finite–dimensional parameter optimiza- tion problem, see e.g. (Hargraves and Paris, 1987; Betts, 1998, 2001; Seywald, 1994; Nocedal and Wright, 1999). The well–known advantage of the direct approach is the good numerical robustness with respect to the initial guess as well as the handling of constraints. On the other hand, indirect approaches are based on Pontryagin’s maximum principle (Pontryagin et al., 1962; Bryson and Ho, 1969) and require the solution of a two–point boundary value problem (BVP). Indirect methods are known to show a fast numerical convergence in the neighborhood of the optimal solution and to deliver highly accurate solutions, which makes them particularly attractive in aerospace industries. However, a main difficulty in the indirect method is the requirement of a good initial guess of the trajectories, especially of the adjoint states. If the initial guess is too far away from the optimal solution, the numerical solution of the BVP will in general fail to converge. The problem of finding a suitable initial guess for the adjoint states has attracted much attention. In particular, von Stryk and Bulirsch (1992) proposed to use both direct and indirect methods combined in a hybrid scheme to overcome the initial guess problem and applied the method to the reentry problem. Further approaches to calculate the adjoint states based on trajectories obtained from direct methods are proposed e.g. by Martell and Lawton (1995) and Seywald and Kumar (1996). However, all these approaches still require the direct method to obtain initial near–optimal guesses for the indirect solution. The main contribution of this paper is to present a new homotopy approach, which is based on an auxiliary OCP for which the adjoint states are simply zero. Starting from the auxiliary OCP, a continuation method is employed to smoothly reach the original OCP. The auxiliary OCP can be derived for any given initial trajectory of the system, e.g. obtained by forward integration. Hence, the homotopy approach can be seen as “self–contained”, since it does not require any initial near–optimal trajectory from direct optimization approaches. For the sake of illustration, the homotopy approach is applied to the space shuttle reentry problem, which is a frequently used benchmark in optimal control due to several challenging features like highly nonlinear dynamics and a high numerical sensitivity with respect to the initial guess of the trajectories. Direct optimization methods have been used for various reentry problems by Betts (2001); Bonnans and Launay (1998), and Neckel et al. (2003) in the context of inverse dynamic optimization. The indirect method based on Pontryagin’s maximum principle has been applied to reentry problems e.g. by Pesch (1994); Kreim et al. (1996); Bonnard et al. (2003). The paper is organized as follows: In Section 2, the homo- topy approach based on the auxiliary OCP is introduced for a general class of OCPs. The reentry problem and the optimal control objective is described in Section 3. Finally, Section 4 shortly explains the collocation method, which is used to solve the 2–point BVP of the reentry problem and presents the reentry trajectories by using the homotopy approach via the auxiliary OCP. 2. HOMOTOPY APPROACH WITH AUXILIARY OCP The considered OCP is to minimize a cost functional J (x,u,t)= ϕ(x(T ),T )+  T 0 L(x,u,t)dt (1) Proceedings of the 17th World Congress The International Federation of Automatic Control Seoul, Korea, July 6-11, 2008 978-1-1234-7890-2/08/$20.00 © 2008 IFAC 14307 10.3182/20080706-5-KR-1001.0638